The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

Rewrite Strategy: INNERMOST

(1) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 4.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4]
transitions:
c0() → 0
n__f0(0) → 0
n__true0() → 0
false0() → 0
f0(0) → 1
if0(0, 0, 0) → 2
true0() → 3
activate0(0) → 4
c1() → 5
n__true1() → 7
n__f1(7) → 6
if1(0, 5, 6) → 1
activate1(0) → 2
n__f1(0) → 1
n__true1() → 3
activate1(0) → 8
f1(8) → 4
true1() → 4
c2() → 9
n__true2() → 11
n__f2(11) → 10
if2(8, 9, 10) → 4
activate1(6) → 1
n__f2(8) → 4
n__true2() → 4
f1(8) → 2
f1(8) → 8
true1() → 2
true1() → 8
if2(8, 9, 10) → 2
if2(8, 9, 10) → 8
n__f2(8) → 2
n__f2(8) → 8
n__true2() → 2
n__true2() → 8
activate2(7) → 12
f2(12) → 1
activate1(10) → 4
activate1(10) → 2
activate1(10) → 8
activate2(11) → 12
f2(12) → 4
c3() → 13
n__true3() → 15
n__f3(15) → 14
if3(12, 13, 14) → 1
n__f3(12) → 1
true2() → 12
f2(12) → 2
f2(12) → 8
if3(12, 13, 14) → 4
n__f3(12) → 4
true3() → 12
n__true3() → 12
if3(12, 13, 14) → 2
if3(12, 13, 14) → 8
n__f3(12) → 2
n__f3(12) → 8
n__true4() → 12
0 → 4
0 → 2
0 → 8
6 → 1
9 → 4
9 → 2
9 → 8
10 → 4
10 → 2
10 → 8
7 → 12
11 → 12
13 → 1
13 → 4
13 → 2
13 → 8

(2) BOUNDS(1, n^1)

(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0) → if(z0, c, n__f(n__true))
f(z0) → n__f(z0)
if(true, z0, z1) → z0
if(false, z0, z1) → activate(z1)
true → n__true
activate(n__f(z0)) → f(activate(z0))
activate(n__true) → true
activate(z0) → z0

Tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))
F(z0) → c2
IF(true, z0, z1) → c3
IF(false, z0, z1) → c4(ACTIVATE(z1))
TRUE → c5
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__true) → c7(TRUE)
ACTIVATE(z0) → c8

S tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))
F(z0) → c2
IF(true, z0, z1) → c3
IF(false, z0, z1) → c4(ACTIVATE(z1))
TRUE → c5
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__true) → c7(TRUE)
ACTIVATE(z0) → c8

K tuples:none
Defined Rule Symbols:

f, if, true, activate

Defined Pair Symbols:

F, IF, TRUE, ACTIVATE

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 5 trailing nodes:

IF(true, z0, z1) → c3
ACTIVATE(z0) → c8
TRUE → c5
ACTIVATE(n__true) → c7(TRUE)
F(z0) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0) → if(z0, c, n__f(n__true))
f(z0) → n__f(z0)
if(true, z0, z1) → z0
if(false, z0, z1) → activate(z1)
true → n__true
activate(n__f(z0)) → f(activate(z0))
activate(n__true) → true
activate(z0) → z0

Tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))
IF(false, z0, z1) → c4(ACTIVATE(z1))
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))

S tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))
IF(false, z0, z1) → c4(ACTIVATE(z1))
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

f, if, true, activate

Defined Pair Symbols:

F, IF, ACTIVATE

Compound Symbols:

c1, c4, c6

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

if(true, z0, z1) → z0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__f(z0)) → f(activate(z0))
activate(n__true) → true
activate(z0) → z0
f(z0) → if(z0, c, n__f(n__true))
f(z0) → n__f(z0)
true → n__true
if(false, z0, z1) → activate(z1)

Tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))
IF(false, z0, z1) → c4(ACTIVATE(z1))
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))

S tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))
IF(false, z0, z1) → c4(ACTIVATE(z1))
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

activate, f, true, if

Defined Pair Symbols:

F, IF, ACTIVATE

Compound Symbols:

c1, c4, c6

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

IF(false, z0, z1) → c4(ACTIVATE(z1))

We considered the (Usable) Rules:

true → n__true
f(z0) → if(z0, c, n__f(n__true))
f(z0) → n__f(z0)
if(false, z0, z1) → activate(z1)
activate(z0) → z0
activate(n__f(z0)) → f(activate(z0))
activate(n__true) → true

And the Tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))
IF(false, z0, z1) → c4(ACTIVATE(z1))
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x₁)) = x₁²   
POL(F(x₁)) = [1] + [2]x₁   
POL(IF(x₁, x₂, x₃)) = [2]x₁ + [2]x₂ + x₃² + [2]x₂·x₃ + x₂²   
POL(activate(x₁)) = x₁   
POL(c) = 0   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(f(x₁)) = [1] + x₁   
POL(false) = [1]   
POL(if(x₁, x₂, x₃)) = x₂ + x₃ + x₂·x₃ + [2]x₁·x₂   
POL(n__f(x₁)) = [1] + x₁   
POL(n__true) = 0   
POL(true) = 0   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__f(z0)) → f(activate(z0))
activate(n__true) → true
activate(z0) → z0
f(z0) → if(z0, c, n__f(n__true))
f(z0) → n__f(z0)
true → n__true
if(false, z0, z1) → activate(z1)

Tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))
IF(false, z0, z1) → c4(ACTIVATE(z1))
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))

S tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))

K tuples:

IF(false, z0, z1) → c4(ACTIVATE(z1))

Defined Rule Symbols:

activate, f, true, if

Defined Pair Symbols:

F, IF, ACTIVATE

Compound Symbols:

c1, c4, c6

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))

We considered the (Usable) Rules:

true → n__true
f(z0) → if(z0, c, n__f(n__true))
f(z0) → n__f(z0)
if(false, z0, z1) → activate(z1)
activate(z0) → z0
activate(n__f(z0)) → f(activate(z0))
activate(n__true) → true

And the Tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))
IF(false, z0, z1) → c4(ACTIVATE(z1))
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x₁)) = x₁ + x₁²   
POL(F(x₁)) = [1] + x₁   
POL(IF(x₁, x₂, x₃)) = [2]x₂ + x₃² + x₂·x₃ + x₁·x₃ + x₁·x₂   
POL(activate(x₁)) = [2]x₁   
POL(c) = 0   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(f(x₁)) = [2] + x₁   
POL(false) = [2]   
POL(if(x₁, x₂, x₃)) = x₁ + [2]x₃ + x₂·x₃ + x₁·x₂   
POL(n__f(x₁)) = [1] + x₁   
POL(n__true) = 0   
POL(true) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__f(z0)) → f(activate(z0))
activate(n__true) → true
activate(z0) → z0
f(z0) → if(z0, c, n__f(n__true))
f(z0) → n__f(z0)
true → n__true
if(false, z0, z1) → activate(z1)

Tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))
IF(false, z0, z1) → c4(ACTIVATE(z1))
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))

S tuples:

F(z0) → c1(IF(z0, c, n__f(n__true)))

K tuples:

IF(false, z0, z1) → c4(ACTIVATE(z1))
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))

Defined Rule Symbols:

activate, f, true, if

Defined Pair Symbols:

F, IF, ACTIVATE

Compound Symbols:

c1, c4, c6

(13) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F(z0) → c1(IF(z0, c, n__f(n__true)))
IF(false, z0, z1) → c4(ACTIVATE(z1))

Now S is empty

(14) BOUNDS(1, 1)